Differentiating acceleration using constant of motion

Let's say we have an object falling through an accelerated field from r to s. I would say a gravitational field, as it is by the same process as gravity, but I will be applying a different constant of motion within the field.

If the constant of motion is something like (1 - b / r) = (1 - (v_r/c)^2), or (1 - b / r) / (1 - (v_r/c)^2) = K (constant for all r), where b is also constant, then the acceleration in terms of r can be found with

b / r = (v_r/c)^2, b / s = (v_s/c)^2

a = d(v^2) / (2 dr)

a = (v_s^2 - v_r^2) / (2 dr), where r - s = dr

a = c^2 (b / s - b / r) / (2 (r - s))

a = c^2 b (r - s) / (2 r s (r - s))

a = c^2 b / (2 r^2)

But what I want to know is how we would find the acceleration similarly to the above example with a constant of motion within an accelerated field of (1 - b / r) = (1 - v_r / c) / (1 + v_r / c) instead?